1 Frequency Modulation (FM)

Transcription

1 Frequency Modulation (FM). Objective This experiment deals with the basic performance of Frequency Modulation (F M). Upon completion of the experiment, the student will: Understand ANGLE modulation concept. Learn how to generate F M signal. Learn how to build F M demodulator. Become familiar with Bessel Function..2 Prelab Exercise..Find the maximum frequency deviation of the following signal; and verify your results in the laboratory. Carrier sinewave frequency 0 M Hz, amplitude V p-p with frequency deviation constant 0.7 khz=v,modulated by sinewave frequency 0 khz amplitude V p-p. 2. Explain what is Carson s rule. 3. What is the di erence between NBFM and wideband FM refer to the Spectral component of the two signals. 4. Use Matlab to draw an FM signal: = 5Hz, carrier amplitude A = 2:5V, A m = V p, modulation frequency f m = Hz, modulator constant K f = 7:5Hz=V olt, t = 0 to 4 seconds. Show : (a) Modulation frequency versus time. (b) FM signal. (c) Di erentiated FM signal. (d) Di erentiated FM signal followed by a LP F. 5. Write a mathematical expression for Fourier transform of a di erentiation, add a graph for the absolute value of the transform, explain the use of a di erentiation in a FM detector.

2 .3 Background Theory An angle modulated signal, also referred to as an exponentially modulated signal, has the form S m (t) = A cos[!t + (t)] = RefA exp[j(t)]g () were the instantaneous phase i (t) is de ned as i (t) =!t + (t) (2) and the instantaneous frequency of the modulated signal is de ned as! i (t) = d d((t)) [!t + (t)] =! + dt dt The functions (t) and d((t)) are referred to as the instantaneous phase dt and frequency deviations, respectively. The phase deviation of the carrier (t) is related to the baseband message signal s(t). Depending on the nature of the relationship between (t) and s(t) we have di erent forms of angle modulation. d((t)) dt (t) = k f Z t (3) = k f s(t) (4) t 0 s()d +!t (5) where k f is a frequency deviation constant, (expressed in (radian/sec)/volt). It is usually assumed that t 0 = and ( ) = 0: Combining Equations-4 and 5 with Equation-, we can express the frequency modulated signal as S m (t) = A cos[!t + k f Z t s()d] (6) Fig. shows a single tone (s (t) message signal), frequency modulated a carrier frequency, represented in time domain. 2

3 mesage signal Carrier FM Fig : modulation signal (upper), carrier signal (midle), and modulated FM signal(lower).4 Bessel Function Bessel function of the rst kind, is a solution of the di erential equation 2 d 2 y dx 2 + dy dx + (2 n 2 )y() = 0 j 0 ÝKÞ Amplitude(volt) j ÝKÞ j 2 ÝKÞ K f modulation index(beta) Fig. 2 Bessel function, of kind, and order to 0 3

4 Bessel function de ned for negative and positive real integers. It can be shown that for integer values of n j n () = ( ) n j n () (7) j n () + j n+ () = 2n j n() (8) X n= j 2 n() = (9) A short listing of Bessel function of rst kind of order n and discrete value of argument ; is shown in Table-, and graph of the function, is shown in Fig. 2. Note that for very small ;value j 0 () approaches unity, while j () to j n () approach zero. nn Table- Bessel function j n ().5 Properties of Bessel function. Eq. -.9 indicates that the phase relationship between the sideband components is such that the odd-order lower sidebands are reversed in phase. 2. The number of signi cant spectral components is a function of argument (see Table-). When, only J 0, and J, are signi cant so 4

5 that the spectrum will consist of carrier plus two sideband components, just like an AM spectrum with the exception of the phase reversal of the lower sideband component. 3. A large value of implies a large bandwidth since there will be many signi cant sideband components. 4. Transmission bandwidth of 98% of power always occur after n = +, we note it in table- with underline. 5. Carrier and sidebands null many times at special values of see table-2. Order for st zero for 2nd zero for 3rd zero for 4th zero for 5th zero for 6th zero Table-2 Zeroes of Bessel function: Values for when j n () = 0.6 Spectrum of Frequency Modulated Signal Since frequency modulation is a nonlinear process, an exact description of the spectrum of an frequency-modulated signal for an arbitrary message signal is more complicated than linear process. However if s(t) is sinusoidal, then the instantaneous frequency deviation of the angle-modulated signal is sinusoidal and the spectrum can be relatively easy to obtained. If we assume s(t) to be sinusoidal then s(t) = A m cos! m t (0) then the instantaneous phase deviation of the modulated signal is (t) = k fa m! m sin! m t () The modulated signal, for the (FM signal), is given by S m (t) = A cos(!t + sin!t) (2) 5

6 where the parameter is called the modulation index de ned as = k fa m! m For FM The parameter is de ned only for sinewave modulation and it represents the maximum phase deviation produced by the modulating signal. If we want to compute the spectrum of S m (t) given in Equation, we can express S m (t) as S m (t) = RefA exp(j!t) exp(j sin! m t)g (3) In the preceding expression, exp(j sin! m t) is periodic with a period T m = 2! m. Thus, we can represent it in a Fourier series of the form exp(j sin! m t) = X C x (nf m ) exp(j2nf) (4) Where C x (nf m ) =! Z m 2 = 2! M! M Z exp(j sin! m t) exp( j! m t)dt (5) exp[j( sin n)]d = j n () Where j n () known as Bessel functions. Combining Equations.3,.4, and.5, we can obtain the following expression for the F M signal with tone modulation: X S m (t) = A j n () cos[(! + n! m )t] (6) The spectrum of S m (t) is a spectrum of a sinusoidal signal multiplied by a constant (Bessel function ), such a spectrum consist of in nite number of Dirac delta function. The number of signi cant (energy contained) spectral lines, is limited, so we can use the FM nodulation, with nite bandwidth. An example of narrow band and wideband FM spectrum is shown in Figure- 3. 6

7 J 0 (0.4) Amplitude(volt) J - (0.4) J (0.4) 0.0 K = 0. 4 K = 5 Amplitude(volt) J 0 (5) J -3 (5) J 3 (5) 0.00 f c -f m f c f f c +f m Fig 3 : Narrow, and wideband FM spectrum. The spectrum of an F M signal has several important properties: 0.00 f c -3f m f c f c +3f m. The F M spectrum consists of a carrier component plus an in nite number of sideband components at frequencies f nf m (n = ; 2; 3:::). But the number of signi cant sidebands depend primarily on the value of. In comparison, the spectrum of an AM signal with tone modulation has only three spectral components (at frequencies f, f + f m, and f f m ). 2. The relative amplitude of the spectral components of an F M signal depend on the values of j n (). The relative amplitude of the carrier depends on j 0 () and its value depends on the modulating signal (unlike AM modulation where the amplitude of the carrier does not depend on the value of the modulating signal)..7 Power and Bandwidth of FM Signals In the previous section we saw that a single tone modulated F M signal has an in nite number of sideband components and hence the F M spectrum seems to have in nite spectrum. Fortunately, it turns out that for any a large portion of the power is contained in nite bandwidth.. Hence the determination of F M transmission bandwidth depends to the question of how many signi cant sidebands need to be included for transmission, if the distortion is to be within certain limits. To determine F M transmission bandwidth, let us analyze the power ratio S n ; which is the fraction of the power contained in the carrier plus n 7

8 sidebands, to the total power of FM signal. We search a value of number of sidebands n; for power ratio S n 0:98: S n = P n j 2 k 2() k= n P j 2 k 2() k= (7) S n 0:98 (8) Using the value properties of Bessel function and Table, we can show that the bandwidth of FM signal B T, depends on the number of sidebands n, and FM modulation index : which can be expressed as.8 Narrow Band FM B T 2( + )f m (9) Narrowband F M is in many ways similar to DSB or AM signals. By way of illustration let us consider the NBF M signal S m (t) = A cos[!t + (t)] = A cos!t cos (t) A sin!t sin (t) t A cos!t A(t) sin!t (20) Using the approximations cos = and sin, when is very small. Equation-26 shows that a NBF M signal contains a carrier component and a quadrature carrier linearly modulated by (a function of) the baseband signal. Since s(t) is assumed to be bandlimited to f m therefore (t) is also bandlimited to f m,. Hence, the bandwidth of NBF M is 2f m, and the NBF M signal has the same bandwidth as an AM signal..9 Narrow Band FM Modulator According to Equation-.20, it is possible to generate N BF M signal using a system such as the one shown in Fig-4. The signal is integrated prior to modulation and a DSB modulator is used to generate the quadrature component of the N BF M signal. The carrier is added to the quadrature component to generate an approximation to a true NBF M signal. 8

9 S(t) φ(t) DSB Aφ(t)sinwt Integrator Modulator Acoswt 90 Shift + NBFM NBFM Modulator Fig 4 NBFM Modulator.0 Wide Band FM Modulator There are two basic methods for generating F M signals known as direct and indirect methods. The direct method makes use of a device called voltage controlled oscillator (V CO) whose oscillation frequency depends linearly on the modulation voltage. A system that can be used for generating an F M signal is shown in Figure-5. S(t) Voltage Control Oscillator Acos[wt+φ(t)] Wideband FM modulator Fig. 5 VCO as wideband FM modulator The combination of message di erentiation that drive a V CO produces a P M signal. The physical device that generates the F M signal is the V CO whose output frequency depends directly on the applied control voltage of the message signal. V CO0s are easily implemented up to microwave frequencies.. Demodulation of FM Signals An F M demodulator is required to produce an output voltage that is linearly proportional to the input frequency variation. One way to realize the 9

10 requirement, is to use discriminators- devices which distinguish one frequency from another, by converting frequency variations into amplitude variations. The resulting amplitude changes are detected by an envelope detector, just as done by AM detector. the discriminator output will be S m (t) = A cos[!t + k f Z t s()d] y d (t) = k d k f s(t) where k d is the discriminator constant. The characteristics of an ideal discriminator are shown in Fig. 6. Discriminator can be realized by using a lter in the stopband region, in a linear range, assuming that the lter is di erentiation in frequency domain. Vout FM signal time, nsec Differentiator d dt Vout time, nsec Envelope Detector yd Differentiator and BPF as FM Discriminator Fig 6 Ideal and real frequency demodulator Vo time, usec Output Voltage Slope kd Frequency Linear range An approximation to the ideal discriminator characteristics can be obtained by the use of a di erentiation followed by an envelope detector (see Figure-6). If the input to the di erentiator is S m (t), then the output of the di erentiator is S 0 m(t) = A[! + k f s(t)] sin[!t + (t)] (2) With the exception of the phase deviation (t), The output of the di erentiator is both amplitude and frequency modulated. Hence envelope detection can be used to recover the message signal. The baseband signal is recovered without any distortion if Maxfk f s(t)g = 2f <!; which is easily achieved in most practical systems. 0

11 2 Experiment Procedure 2. Required Equipment. Spectrum Analyzer (SA) HP 8590L. or equivalent 2. Oscilloscope HP 54600A. 3. Signal Generator (SG) HP=Agilent 8647A. 4. Function Generator HP 3320A. 5. Double Balanced Mixer Mini-Circuit ZAD Phase Shifter Mini-Circuit ZSCQ MHz band pass lter. Mini Circuit BBP VCO as Frequency Modulator 2.2. Simulation In this simulation we drive the VCO (FM modulator) by DC and AC signal, and record the output signal.. Simulate a VCO as FM modulator as indicated in Fig. 7

12 TRANSIENT Tran Tran StopTime=200.0 usec MaxTimeStep=0.0 nsec VtSine SRC Vdc=V Amplitude=0 V Freq=0 khz Delay=0 nsec Damping=0 Phase=0 f VspecTran VspecTran VspecTran VspecTran=vspec_tran(Vout,0,0kHz,50) Vout R R VCO R=50 Ohm VCO Kv=50.0 khz Freq=200 khz P=-j*dbmtow(0) Rout=50 Ohm Delay=timestep Harmonics=list(-0.0, 0.002) Fig. 7 VCO as FM modulator 2. What is the expected frequency of V out signal? Draw a graph of the signal in time domain and frequency domain and prove your answer.. 3. Set the voltage of the modulating frequency to V dc = 0 and Amplitude to V. 4. Draw a graph of FM signal in frequency and time domain. 5. Calculate modulation index of the modulated signal;and use MAT- LAB command J = besselj(0 : 6; 5) to nd the amplitude of sidebabands components. compare the results to the simulation VCO as Frequency Modulator- Measurement During this experiment you learn how to measure the F M modulation characteristics and Bessel function in frequency domain using spectrum analyzer... Connect the system as indicated in Fig Set the Signal-generator to: frequency 0 M Hz, amplitude 0 dbm. External DC F M modulation, frequency deviation 20 khz. 2

13 3. Set the function generator to DC volt, amplitude 530 mv. Signal Generator ,00 MHz Spectrum analyzer LPF 5.000,000 MHz Function Generator Fig 8 VCO as FM modulator 4. Adjust the amplitude (if necessary) of the function generator in order to get full frequency deviation. 5. Measure and record the frequency of the signal generator with spectrum analyzer. 6. Change the DC voltage, and watch the output signal, Switch o the modulation, measure and record the frequency of the signal generator. 7. Calculate the VCO Frequency tuning sensitivity (Hz/Volt) K V Frequency Modulation and Bessel Function. Connect the function generator directly to the spectrum analyzer as indicated in Fig. 9 Function Generator 5.000,000 MHz Spectrum analyzer LPF Fig. 9 Frequency modulation and Bessel function. 3

14 2. Set the function generator to sinewave frequency 0MHz, amplitude V RMS, FM, modulating frequency 5 khz, frequency deviation 25kHz. 3. Calculate the modulation index ; switch o the modulation, and verify by the spectrum analyzer that the amplitude of the carrier only is exactly V RMS : 4. Switch on the FM modulation, and measure the amplitude, and frequency of the sidebands is according to Bessel function, see table-. Save the image on magnetic media. 5. Adjust the function generator to proper frequency deviations to get rst null of the carrier j 0 = 0; and rst null of the sideband j = 0; Save the image on magnetic media. 2.3 Power and Bandwidth of FM Signal-Carson s Rule 2.3. Simulation. Simulate a frequency modulated signal, as indicated in Fig. 0 TRANSIENT Tran Tran StopTime=200.0 usec MaxTimeStep=0.0 nsec VtSine SRC Vdc=0 V Amplitude= V Freq=0 khz Delay=0 nsec Damping=0 Phase=0 f PspecTran I_Probe VCO I_Probe VCO Kv=20 khz Freq=200 khz P=-j*dbmtow(3) Rout=50 Ohm Delay=timestep Harmonics=list(-0.0, 0.002) PspecTran PspecTran2 PspecTran2=pspec_tran(Vout, 0,I_Probe.i,0kHz,50) Vout R R R=50 Ohm Fig. 0 Simulation of power and bandwidth of FM signal 4

15 2. According to Carson s rule nd the bandwidth and power of the FM signal, 3. Switch o the FM modulator (set the amplitude of VtSine to 0V), measure the total power of the signal and show that, at least 98% of power contained in Carson s bandwidth Measurement of Power and Bandwidth of FM Signal. Connect the function generator to the spectrum analyzer as indicated in Fig Set the function generator to sinewave frequency 0MHz, amplitude V RMS, FM, modulating frequency 5 khz, frequency deviation 0kHz. 3. Find the bandwidth of the signal according to Carson s rule, and show such a bandwidth contains at least 98% of total power. (total power is the power of the signal without modulation). 2.4 Narrow Band FM Modulator In this part of the experiment, you implement narrow FM modulator (see Fig. 4), without the rst stage- integrator, since we assume that our sinewave modulating signal, is a phase shifted of other sinewave signal Simulation. Simulate a narrow band modulator (see Fig. 4) use the elements of Fig.. 5

16 Fig.- Narrow band FM signal (a) Carrier-sinewave frequency 0MHz amplitude 2V p : (b) modulating frequency- sinewave frequency 00kHz, amplitude 40mV. 2. According to Carson s rule nd the bandwidth of the FM signal. Draw a frequency domain graph signal Magnitude (voltage) versus frequency, and nd the voltage of the carrier and sidebands? Use Table- to nd the modulation index Narrow Band FM-Measurement. Connect the system according to Fig Adjust the equipment as follow: function generator-lo - Sinewave frequency 0 MHz amplitude 7dbm. function generator- RF- Sinewave frequency 0 khz amplitude -0dbm.(integral of the cosine input wave). 6

17 Function Generator 5.000, 000 MHz Spectrum analyzer Power 2 splitter s LPF Function Generator 5.000, 000 MHz s 9 0 E phase shift 2 Fig.-2 Narrow band FM modulator. 3. Set the spectrum analyzer to 0 MHz span 50kHz, watch the FM signal at spectrum analyzer, change the amplitude and frequency of the modulating frequency generator, which component of the FM signal changed? nn (Ref.) 0.00(Ref) -26.0(dB) -22.5(dB) Table-3 NBFM, sidebands amplitude 4. Change the amplitude and frequency of the local oscillator, which component of the FM signal changed? 5. According to table-3 set the system to = 0:; 0:5, calculate the frequency deviation for each, save image on magnetic media. 2.5 FM Demodulator- discrimination Method Simulation In this part you will demodulate an FM signal using a discriminator. A discriminator may be realized by a LPF (di erentor in frequency domain) followed by an envelope detector.. Simulate an FM demodulator based on discriminator as indicated in Fig. 3 7

18 Diode_Model DIODEM Is= Bv= Rs= Ibv= Gleak= Nbv= N= Ibvl= Tt= Nbvl= Cd= Kf= Cjo= Af= Vj= Ffe= M= Jsw= Fc= Rsw= Imax= Gleaksw = Imelt= Ns= Isr= Ikp= Nr= Cjsw= Ikf= Msw= Vjsw= Fcsw= AllowScaling=no Tnom= Trise= Xti= Eg= AllParams= TRANSIENT Tran Tran StopTime=80.0 usec MaxTimeStep=0.50 nsec R R2 R=50 Ohm LPF_Gaussian LPF Fpass=2 MHz VAM Apass=3.003 db VtSFFM GDpass=0.9 SRC3 Vdc=0 V Amplitude=3 V CarrierFreq=2.3 MHz ModIndex=2.5 SignalFreq=25 khz Vout Diode DIODE Model=DIODEM Area= Periph= Scale= Region= Temp= Trise= Mode=nonlinear PRC PRC R=2 kohm C=5 nf Fig. 3 FM detector- discrimination method 2. Draw a graph of VAM, and Vout, explain the idea of the circuit. 2.6 FM Demodulator 2.6. Measurement We start in the rs part with a low pass lter as discriminator, in the second part we use the IF lter of the spectrum as a discriminator, and peak detector as demodulator.. Connect the lter and function generator to the oscilloscope,as indicated in Fig

19 Oscilloscope Function generator 5.000,000 MHz Function generator 5.000,000 MHz LPF.9MHz Spectrum analyzer LPF Fig. 4 FM Discriminator 2. Set the function generator to FM modulation, carrier frequency 2.4 MHz, Amplitude 0 dbm, modulating frequency k Hz. deviation frequency 50 khz. 3. The oscilloscope display an FM signal after di erentiation, which signal you identify, save image on magnetic media 4. Disconnect the lter and connect the function generator to the spectrum analyzer. 5. Set the spectrun analyzer to center frequency 2.4MHZ, span 300kHz, IF bandwidth to 0kHz. Center the signal on display if necessary. 6. Set the span to 0kHz, now the spectrum display a signal like an oscilloscope, change the center frequency slightly to recover the modulating frequency (khz.). 9